Two Points Moving...

Date: 7/18/96 at 16:58:53
From: Bill Buchanan
Subject: Two Points Moving...
I would appreciate help in solving this problem. If you can help me
please explain the steps involved.
Point P is moving in the positive direction, along the y axis, at a
constant velocity of 10 metres per second. Point Q is moving with a
constant speed of 11 metres per second (relative to the x-y plane) and
always in the direction directly towards point P. If at time t = 0,
P is at the origin and Q is at the point (100,0) what is the equation
of the locus of point Q ?
Thank you for your kind assistance.
Michael Alger

Date: 7/19/96 at 8:25:51
From: Doctor Anthony
Subject: Re: Two Points Moving...
Draw the diagram I describe, and refer to it as we work through the
problem.
We use the x and y axes in the usual way with P moving along OY and Q
starting at (100,0) Let this point be Q0. Because of ratio of speeds,
after time t, the path distance Q0->Q will be 11*t and OP will be
10*t, and if 's' is the distance along Q's path we have s/11 = OP/10
or s = (11/10)*OP
Now the tangent at Q to the path will be pointing directly at P, so if
dy/dx is the slope of the tangent and (x,y) the coordinates of Q we
have
(OP-y)/x = -dy/dx
The minus sign because (OP-y)/x is positive, whereas the slope must be
negative if y is increasing while x is decreasing.
OP - y = -x(dy/dx)
OP = y - x(dy/dx) But s = (11/10)OP, so
s = (11/10){y - x(dy/dx)}
We must try to get a differential equation in x and y, so will have to
get rid of s. Differentiate above equation with respect to x.
ds/dx = (11/10){dy/dx - dy/dx - x(d^2y/dx^2)}
= - (11/10)(x*d^2y/dx^2) .....(1)
But dx/ds is the cos of the angle of slope, so ds/dx = sec of angle of
slope so ds/dx = sqrt(1 + tan^2(angle of slope)) = sqrt(1 + (dy/dx)^2)
In this case because s increases while x decreases we have:
ds/dx = -sqrt(1 + (dy/dx)^2) ......(2)
Now to simplify the working, let p = dy/dx, so dp/dx = d^2y/dx^2
Equating the right hand sides of equations (1) and (2)
sqrt(1+p^2) = (11/10)(x*dp/dx) Separating the variables we get
dx/x = (11/10)(dp/sqrt(1+p^2))
ln(kx) = (11/10)sinh^(-1)p k is constant of integration.
Now x = 100 when p = 0 so ln(100k) = 0, 100k = 1, k = 1/100
ln(x/100) = (11/10)sinh^(-1)p
p = sinh{ln(10/11)ln(x/100)} = sinh{ln(x/100)^(10/11)}
dy/dx = sinh{ln(x/100)^(10/11)}
= (1/2){e^(ln(x/100)^(10/11) - e^(-ln(x/100)^(10/11)}
= (1/2){(x/100)^(10/11) - (x/100)^(-10/11)}
y = (1/2)*100*{(11/21)(x/100)^(21/11) - 11*(x/100)^(1/11)} + const
y = 0 when x = 100 so
0 = (1/2)*100*{(11/21) - 11} + const
= -11000/21 + const, so const = 11000/21
Equation of curve is :
y = 50{(11/21)(x/100)^(21/11) - 11(x/100)^(1/11)} + 11000/21
Q will meet P when x = 0, and this occurs at y = 11000/21 = 523.81
meters from O
-Doctor Anthony, The Math Forum
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